Equidistribution of Heegner points and ternary quadratic forms

Abstract

We prove new equidistribution results for Galois orbits of Heegner points with respect to single reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize an equidistribution result with respect to a single reduction map established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.

Rankin–Selberg L-functions and the reduction of CM elliptic curves

“…Michel [30,Theorem 3] proved a "sparse" equidistribution version of (1.3), where G K can be replaced by any subgroup G < G K of index ≤ D 1/2115 . Related equidistribution problems were studied in [6,22,34].…”

Liftings of reduction maps for quaternion algebras

“…Given such a reduction map, it is expected that the Galois orbits in CM(G, H) tend to be equidistributed among the finitely many fibres of the reduction map red. Such equidistribution results are already known in various cases [4,11,12,14,18] and were crucial in the proof of Mazur's non-vanishing conjecture by the first author and Vatsal [3,5,21].…”

On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring

“…This question was studied by a number of authors (cf. [5] and [10]). It turns out that the reduction map is not always surjective and is not in general one-to-one.…”

“…They proved that for d sufficiently large, the image of the reduction map is surjective and furthermore that it is equidistributed across all supersingular elliptic curves. A slight modification of this was investigated by Jetchev and the second author [10], where it was shown that the reduction from curves with exact CM by O d is surjective for d sufficiently large but not necessarily fundamental (albeit with some minor restriction on the choice of d). The approach taken in [5] and [10] was to use a correspondence between elliptic curves with CM by O d which reduce to a supersingular elliptic curve and optimal embeddings of O d in its endomorphism ring; roughly speaking, if O d embeds into the quaternionic order, then O dr 2 also embeds by multiplying by r, and optimal embeddings are those which do not come from smaller discriminants.…”